Global Harmonic Analysis

PI将证明经典力学和量子力学之间关系的严格结果。薛定谔在1927年引入了量子力学，以解决电子如何同时运动和静止的难题。他的答案是，具有确定能量的量子粒子只能存在于定态中，定态是控制物理学的薛定谔算符的特征函数的物理术语。问题是特征函数很难可视化和理解。整体调和分析就是在这些特征函数和物理系统的经典力学之间建立严格的关系。提案中的项目解决了两种相反的问题：(i)特征函数可以有多大，以及它的大点分布如何？这些是最有可能找到粒子的点。（ii）特征函数为零的点是如何分布的？这些是最不可能找到粒子的点。全局谐波分析是利用黎曼流形测地线流的长时间动力学来理解特征函数的高频渐近性。其中最重要的两个问题是：(1)分析特征函数的Lp范数和(2)分析节点集和临界点集。对于较大的p， Sogge和PI表明，只有当自焦点x存在时，从x到x的测地线的正测度才能得到通用Lp界。在最近的工作中，PI表明，如果度量是实解析的，那么所有测地线都回环，并且第一个返回映射保留了勒贝格测度类中x处单位切空间上的有限测度。在二维空间中，所有通过x的测地线都是平滑闭合的。一个项目是将最后的结果推广到更高的维度和平滑度量。研究低Lp规范也是很有趣的我们正在用Kakeya-Nikodym最大函数来做这件事。关于节点集和临界点集，J. Jung和PI最近证明了具有非空凹边界的非正曲面上的Neumann或Dirichlet特征函数的节点域的数目必须随着特征值趋于无穷。还证明了特征函数有许多临界点。在未来的工作中，PI将证明一个更定量的结果，并研究更高维度的情况。PI还在实解析情况下使用特征函数的全纯扩展，以获得对节点和临界点集的更多控制。

The PI shall prove rigorous results about the relations between classical and quantum mechanics. Schrodinger introduced quantum mechanics in 1927 to solve the puzzle of how an electron can be moving and stationary at the same time. His answer was that a quantum particle with a definite energy can only exist in a stationary state, the physics word for an eigenfunction of the Schrodinger operator governing the physics. The problem is that eigenfunctions are very hard to visualize and understand. Global Harmonic Analysis is about making rigorous relations between these eigenfunctions and the classical mechanics underlying the physical system. The projects in the proposal address two opposite kinds of questions: (i) how large can the eigenfunction be, and how are the points where it is large distributed? These are the points where it is most probable to find the particle. (ii) How are the points where the eigenfunction is zero distributed? These are the points where it is least likely to find the particle. Global Harmonic Analysis is the use of the long-time dynamics of the geodesic flow of a Riemannian manifold to understand the high-frequency asymptotics of eigenfunctions. Two of the most important problems are (1) to analyze Lp norms of eigenfunctions and (2) to analyze nodal sets and critical point sets. For large p, Sogge and the PI have shown that the universal Lp bounds are only obtained if there exists a self-focal point x where a positive measure of geodesics from x loop back to x. In recent work, The PI showed that if the metric is real analytic, then all geodesics loop back and the first return map preserves a finite measure on the unit tangent space at x in the class of Lebesgue measure. In two dimensions, it follows that all geodesics through x are smoothly closed. One project is to generalize the last result to higher dimensions and smooth metrics. It is also very interesting to study low Lp norms and we are working with the Kakeya-Nikodym maximal function to do that. Regarding nodal and critical point sets, J. Jung and the PI have recently shown that the number of nodal domains of Neumann or Dirichlet eigenfunctions on non-positively curved surfaces with non-empty concave boundary must tend to infinity with the eigenvalue. It was also showed that the eigenfunctions have many critical points. In future work, the PI shall prove a more quantitative result and to study the higher dimensional case. The PI is also using holomorphic extensions of eigenfunction in the real analytic case to gain more control over nodal and critical point sets.